Specification of the laser engraving machine

Manufacturer/model: 2000 Laser, Multicam

Laser frequency:  10000 Hz

Laser medium:  CO2

Wavelength:  10.6 µm

Wave mode:  Pulsed

Table 1:The specification of CO2 laser machine
Figure 1: laser ablation process showed the ejected material atoms from the surface and Heat affected zone due to the laser irradiation of the surface
Figure 2(a): Schematic diagram of CO2 Laser engraving and cutting system
Figure 2(b): shows the image of laser beam spot area with the mechanical device designed to modify the focused area of CO2 onto the irradiated textile sample
Figure 2(c): Shows CO2 Laser Engraving and Cutting Machine
Figure 3: SEM Images showed the effect of CO2 incident Laser Power on the surface of five different synthetic Textile samples; three different laser Power values of 1- Low power at 80 watts, 2- Medium power at 100 watts, and 3- High power at 120 watts, with two different magnification values of 100X and 200X

Normally in literature, the 2D X-ray diffraction optics theory has been based on the differential partial-in-derivative Taka gi–Taupin (TT) equations when the fractional-order parame ter α = 1 (see [1]). In the last decades, substantial progress has been achieved in mathematical physics using di erential equations with fractional order derivatives [2]. Cauchy problems for systems of fractional di erential equations, which act as a mathematics basis for various physical models have been studied [3–7]

Following this logic, one can push one step further in the di raction optics theory now founded on the TT-type equations with the fractional derivatives of the arbitrary order α [0, 1] along the direction 0z of the energy ow propagation within a crystal medium. In this paper, based on the technique of double Fourier–Laplace transform, the integral matrix Fredholm–Volterra equation of the second kind is derived, which is equivalent to the two- dimensional Cauchy problem of di ractive optics. e work goal is to develop the integral formalism of the two-dimensional theory of di ractive optics, previously proposed by the authors [10], based on the fraction al Cauchy problem. In the case, when the imperfect crystal displacement eld function f (R) is a linear function of x, namely: f (R) = ax + b, and a, b = const, one nds out an analyt ical solution of the Cauchy problem for an arbitrary fraction al-order parameter (FOP) α, α E (0, 1].

Accordingly, the original system of fractional di raction optics equations takes the form (cf. [7])

with the Cauchy problem’s condition

where,

is the Gerasimov–Caputo fractional derivative beginning at point a (cf. [3]), is the Riemann–Liouville fractional

integro-di erential operator of order ν is equal to

and for ν ≥ 0 the operator can be determined by recursive relation

T(Z) is the Euler gamma-function, φ0 (x) and φh (x) are the given real-valued functions.

Note that in the limit case of the FOP α = 1 the operator reduces to the

Standard derivative

The main point of this paper is to solve the boundary-valued Cauchy's problem in the 2D ’fractional’ X-ray di raction crystal optics taking into account the non-locality of the X-ray-crystal medium interaction.

The paper is organized as follows: Section 1 contains an introductory part, in which the purpose and ideology of the work are explained.

In Section 2, using the method of dou- ble Fourier–Laplace integral transform, Cauchy's problem is reduced to the Fredholm– Volterra integral matrix equation of the second kind.

In Section 3, an explicit solution of Cauchy's problem (1) – (2) is obtained in the case of a linear function f(R). In Section 4, it is shown that in the case of constant initial amplitudes (conditions of Cauchy's problem), this solution is expressed through two-parameter Mittag-Le er functions. In Section 5, conclusions are presented regarding the prospects of the considered theoretical approach to modeling two-dimensional X-ray di raction scattering.

Let us convert the Cauchy problem in the ’di erential derivative' form (1)–(2) to the Fredholm– Volterra-type integral matrix equation of the second kind. e construction of the resolvent of this equation in terms of a Liouville-

In Section 4, it is shown that in the case of constant initial amplitudes (conditions of Cauchy's problem), this solution is expressed through two-parameter Mittag-Le er functions. In Section 5, conclusions are presented regarding the prospects of the considered theoretical approach to modeling two-dimensional X-ray di raction scattering.

Neumann series is of great importance for computer modeling and subsequent reconstruction of the crystal displacement field function f(R) from X-ray di raction microtomography data.

The system of di erential TT-type equations (1) may be rewritten into the form

where

Acting onto both sides of (5) by the operator taking into account that the column vector E = E(x, t) is the Solution of Eq(5) and K2 is equal to Unit matrix one can obtain

Further, we denote the Fourier transform of the function f(x) by (f(x))k , the Laplace transform of the function g(t) by (g(t))p , and respectively, the double Fourier–Laplace transform of the function h(x, t) by (h(x, t))k,p . Using the following formula for the Laplace transform of the fractional derivative

one can get

Keeping in mind Eq. (7), and applying the double

Fourier–Laplace transform to Eq. (6), one obtains

Applying the Efros theorem for operational calculus, the formula

for the Laplace trans- form of the Wright function [2, 3]

the following well-known integrals

the inverse Fourier–Laplace transform for the relation (8) takes the form

where

Θ(x) istheHeavisidefunction,J0(x) isthezero-orderBessel function,

istheWrightfunction(see, e,g., [3]),and

is the Green function introduced in[7].

Taking into account Eqs. (13), (14), the integral matrix equation (12) may be to reduce

where

Thus, according to Eq. (16), the Cauchy problem for the matrix Eq. (1) is reduced to the system of Fredholm-Volterra integral equations.

The unique solution of the matrix integral equation (16) has the form

where

The convergence of series (17) can be easily established from the properties of function (15).

Here we will build up a solution of the basic fractional Cauchy problem when the crystal- lattice displacement eld function f (R) = ax + b.

A er trivial exponentional substitutions for the wave amplitudes E0 (x, t), Eh(x, t), the system (1) can written down as (for simplicity, further, the same notations for the wave amplitudes E0 (x, t), Eh(x, t), are to be saved)

Substituting the functions E0(x,t), Eh(x,t) as

one obtains

and the initial wave field amplitudes condition

Then, let us introduce the notations

O (x) is the Heaviside function, Jm(z) is the m-order Bessel function of the argument z. According to [9], the solution of the Cauchy problem (19) – (20) has the form

in the class of function

Where is the |x| → ∞ function, which satis es the Holder’s condition, and the following relation as

Underline the solution to the Cauchy problem (19) – (20) is unique in the class of functions, which satisfy the condition

From some k > 0 problem (18), (2) can be cast into the form

From Eqs. (20) – (24) it follows the solution of the Cauchy

where the following notations are introduced

Let us consider the case when the initial conditions for the wave eld amplitudes E0(x,0) and Eh(x,0) are constant and equal to

Then, the formulae (25)–(27) can be simpli ed and expressed in terms of the Mittag– Le er-type functions. Keeping in mind Eq. (28), Eq. (25) for E(x, t) can be written down as

By changing the integration order, let us evaluate the integral I1 (x, t).

Let calculate the integrals involved in the representation of the term I1 (x, t). We will need

where

Using formula(32), one can obtain

From the Eqs. (31)–(33), it directly followsup

where Sij (x, x − η, τ ) (i, j = 1, 2) are the elements of matrix S(x, x − η, τ ); and

Next, a er some routine calculations, from (30), (34) and (35) one obtains

where

where a From( 27) one finds out

It is known that following Stankovic’s transformation integral (see [2,3])

takes place for any λ∈C,µ∈(0,1),ν∈R, where

is the Mittag–Le er-type function [8]. Applying formula (38) to equalities (36) and (37), the total solution (29) can be cast into the form

Using the properties of a Mittag-Le er type function, it is easy to show that function (39) provides the proper solution of Cauchy's problem (18). In the case when the FOP α → 1, in view of the relation

the total solution(39) is reduced to (cf. [7])

In this paper, the goal of the study is to elaborate the mathematics model for describing the X-ray propagating via imperfect crystals under the non-locality interaction of the X- ray wave eld with atoms of crystal medium that probably can be important for digital decoding the nm-scale crystal defects in the computer X-ray di raction microtomography (cf., [1]). The Cauchy problem of the 2D fractional X-ray di raction optics designed to describe the mathematical model of the X-ray propagation via the imperfect crystal has been described in terms of the matrix integral Fredholm–Volterra equation. e matrix Resolvent solution of the Cauchy problem in the 2D fractional X-ray di raction optics has been built and analyzed for the case of the coherent two-beam X-ray di raction by imperfect crystals under the non-locality interaction of the X-ray with atoms of crystal medium along the crystal thickness. It is shown that in the case, when the fractional order parameter (FOP) α = 1, the results obtained have been directly suitable to the mathematical model used of the 2D standard X-ray di raction optics used in the computer X-ray di raction microtomography (cf. [7]). Solving the fractional integral-derivative Cauchy problem above presented should be considered as some attempt to take into account the non-locality of the X-photon-atoms interaction in theory of X-ray di raction crystal microtomography of crystals. To be noticed, the further development and improvement of the theory are a good topic for future work.

The authors con rm contribution to the paper as follows: study conception and design: Murat O. Mamchuev, Felix N. Chukhovskii; analysis and interpretation of results: Murat O. Mamchuev, Felix N. Chukhovskii; dra manuscript preparation: Murat O. Mamchuev, Felix N. Chukhovskii. All authors reviewed the results and approved the nal version of the manuscript.

The author(s) received no speci c funding for this study.

This Work was carried out within the framework of the state assignment of the Ministry of Education and Science of the Russian Federation under the project: Investigation of boundary value problems for equations with generalised fractional di erentiation operators and their application to mathematical modelling of physical and socio-economic processes (1021032424223-6).

The authors declare that they have no con icts of interest to report regarding the present study.

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